Er... this is getting sillier by the minute. Even if the value of something is "1", it has DIMENSIONS. 1 m/s is not equal to 1 potato. But in your scheme of things, it is, because you mix and match the dimensions no matter what.

If you ignore the dimensions of the things you are manipulating, then you can get all kinds of nonsense. That is why this is physics, and not mindless mathematics.

therefore the equation holds true only as long as the base unit of time remains unchanged

so any change to the second would require a revised equation

otherwise we might was well say a = p / ud

where p = a potato, u = an untidy cat, and d = a picture of a happy dog

This is why we teach kids in school to solve the problems algebraically and NOT plug in numbers into the problem until the very end.

The units for an 'acceleration' is defined as Length/(Time)^2. So now, let's look at what you did:

then a = 1 m/s² since v = 1 and t = 1

therefore 1 = 1/1 or 1 = 1*1 or 1 = 1*1² or 1 = 1*1³

so a = v/t or a = v*t or a = v*t² or a = v*t³

What is "1 = 1*1"? Did you do v = at? Fine, but 1 = 1*1^2? If what you did here is

v = a*t*t, then the equality NO LONGER HOLDS! Again, this is blind mathematics. These symbols have PHYSICAL MEANINGS that you simply cannot multiply at will. This is why I said why we teach students to do physics problems algebraically. By saying v = at^2, you have already commited yourself to changing the definition of what a "velocity" is that isn't the same as before.

I think that you are working backwards, Gregor. You cannot base a physical claim on a mathematical derivation that does not look at a system from a physical point of view. Zapperz is quite right in saying that there is no physical meaning in what has been described. By writing your new equations involving t by the observation of the fact that the "equality holds true" from a mathematical standpoint when multiplied by 1, you are changing the units of your system that no longer mean anything to what you are trying to prove (velocity). Indeed, this is an important difference between math and physics. For example, math would probably describe sqrt(2) as the irrational number that gives 2 when squared (gross simplification and/or generality, but lets keep this simple). In math, sqrt(2) can even be negative! Physics would probably see sqrt(2) as the measure of the hypotenuse of a right triangle whose adjacent sides measure 1. The important word here is MEASURE. This implies units. In math, sqrt(2) has no units. Dimensional analysis pretty much seperates math and physics and one can easily poison either discipline by forgetting that. In a way, perhaps i should ask that you avoid doing this as you would to divide by 0 (gross simplification). Allow me to divide by 0, and i can prove anything you wish (it just wont make any sense!) I am not saying that you cannot use math in physics, quite the opposite! But you must be wary of your logic. Physics requires a certain thought process that is similar (but not identical) to mathematics.

There is a difference in doing pure number theory and manipulation versus something with a physical quantity. If you wish to do the latter, you shouldn't have done it in the PHYSICS section of the forum.

And just so you know, in many areas of physics, h=k=c=1 by "definition". But this isn't a new definition. It is for "simplicity" and people in the field know what is being done so there's no confusion. It is why you see people often quoting the mass of a particle simply in terms of MeV rather than MeV/c^2. We all know how to convert those back to their true units.

Moral of the story: please do mathematics in the mathematics forum, especially if you wish to neglect the physical dimension of the quantities you are manipulating. If you wish to carry the discussion in the physics forums, then all the physics rules apply.

but surely you're not suggesting that SI units are universal constants,
carved in stone. in many cases equations are only approximations.
(such as mc²)

It has nothing to do with "units". It has everything to do with dimensions. But even then, you cannot add something with units of meters to something with units of microns. You'll get nonsense.

if a unit changes then it's correlation to other units also changes
and this effects the values in dimensional analysis.

in the example...

does a = v/t still hold true through time in dimensional analysis?

(assuming that tomorrow the BIPM decided to redefine the meter and second)

this is not a problem for the maths forum
it's a physics problem

I would hope this question gets more than a sweeping dismissal, and lengthy explaination as to why such questions are not acceptable in Physics.

a = v/t is the DEFINITION when a is a constant. It is how it is DEFINED regardless of the units involved. The DIMENSION remains no matter what.

And note that you were the one who said that you were more interested in the manipulation of the mathematics. If you notice, you did something that was physically nonsensical, but you justified it simply by the fact that you're not doing anything wrong numerically. That is what prompted me to request that if you wish only to care about the numerical manipulation of numbers with utter disregard to the physical significance of that number, then the physics forum is NOT the place to do it. I'm only going by YOUR intentions.